Define a new operation * on the set of all natural numbers by m n* = m² + n².

(i) Is N closed under ?

(ii) Is commutative on N?

(iii) Is * associative on N?

(iv) Is there an identity element in N with respect to *?

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Solution

(i) Since the sum of the squares of positive integers is a positive integer, N is closed under .

(ii)

Thus, the operation is commutative on N.

(iii) Since,

Thus, the operation * is associative on N.

(iv) Consider .

This is possible only when k = 0.

However, 0 is not a natural number.

Therefore, there is no identity element in N with respect to *.

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Define a new operation * on the set of all natural numbers by m * n = m2 + n2. (i) Is N closed under *? (ii) Is * commutative on N? (iii) Is * associative on N? (iv) Is there an identity element in N with respect to *?

Define a new operation * on the set of all natural numbers by m n* = m² + n².

(i) Is N closed under ?

(ii) Is commutative on N?

(iii) Is * associative on N?

(iv) Is there an identity element in N with respect to *?